scholarly journals On the Frequency of Algebraic Brauer Classes on Certain Log K3 Surfaces

2018 ◽  
Vol 62 (3) ◽  
pp. 551-563
Author(s):  
Jörg Jahnel ◽  
Damaris Schindler
Keyword(s):  
K3 Surfaces ◽  
Upper Bounds ◽  
Hasse Principle ◽  
Brauer Group ◽  
Integral Points ◽  
Quadratic Equations

AbstractGiven systems of two (inhomogeneous) quadratic equations in four variables, it is known that the Hasse principle for integral points may fail. Sometimes this failure can be explained by some integral Brauer–Manin obstruction. We study the existence of a non-trivial algebraic part of the Brauer group for a family of such systems and show that the failure of the integral Hasse principle due to an algebraic Brauer–Manin obstruction is rare, as for a generic choice of a system the algebraic part of the Brauer-group is trivial. We use resolvent constructions to give quantitative upper bounds on the number of exceptions.

scholarly journals Lifting, restricting and sifting integral points on affine homogeneous varieties

2012 ◽  
Vol 148 (6) ◽  
pp. 1695-1716 ◽  
Author(s):  
Alexander Gorodnik ◽  
Amos Nevo
Keyword(s):  
Homogeneous Spaces ◽  
Acta Math ◽  
Upper Bounds ◽  
Lattice Points ◽  
List Type ◽  
Integral Points ◽  
Counting Problem ◽  
Symmetric Varieties ◽  
Almost Prime ◽  
Homogeneous Varieties

AbstractIn [Gorodnik and Nevo,Counting lattice points, J. Reine Angew. Math.663(2012), 127–176] an effective solution of the lattice point counting problem in general domains in semisimpleS-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action ofGonG/Γ, and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in [Nevo and Sarnak,Prime and almost prime integral points on principal homogeneous spaces, Acta Math.205(2010), 361–402] and use them to establish several useful consequences of this property, including:(1)effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties;(2)effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties;(3)effective lower bounds on the number of almost prime points on symmetric varieties;(4)effective upper bounds on almost prime solutions of congruences in homogeneous varieties.

scholarly journals Failure of the Hasse principle on general surfaces

2013 ◽  
Vol 12 (4) ◽  
pp. 853-877 ◽  
Author(s):  
Brendan Hassett ◽  
Anthony Várilly-Alvarado
Keyword(s):  
Algebraic Surface ◽  
Hodge Theory ◽  
Double Cover ◽  
Hasse Principle ◽  
Brauer Group

AbstractWe show that transcendental elements of the Brauer group of an algebraic surface can obstruct the Hasse principle. We construct a general $K 3$ surface $X$ of degree $2$ over $ \mathbb{Q} $, together with a 2-torsion Brauer class $\alpha $ that is unramified at every finite prime, but ramifies at real points of $X$. With motivation from Hodge theory, the pair $(X, \alpha )$ is constructed from a double cover of ${ \mathbb{P} }^{2} \times { \mathbb{P} }^{2} $ ramified over a hypersurface of bidegree $(2, 2)$.

scholarly journals Effective Hasse principle for the intersection of two quadrics

2016 ◽  
Vol 19 (A) ◽  
pp. 73-82
Author(s):  
Tony Quertier
Keyword(s):  
Rational Solution ◽  
Hasse Principle ◽  
Explicit Algorithm ◽  
Quadratic Equations ◽  
Smooth System

We consider a smooth system of two homogeneous quadratic equations over $\mathbb{Q}$ in $n\geqslant 13$ variables. In this case, the Hasse principle is known to hold, thanks to the work of Mordell in 1959. The only local obstruction is over $\mathbb{R}$. In this paper, we give an explicit algorithm to decide whether a nonzero rational solution exists and, if so, compute one.

scholarly journals On the Supersingular Reduction of K3 Surfaces with Complex Multiplication

2018 ◽  
Vol 2020 (20) ◽  
pp. 7306-7346
Author(s):  
Kazuhiro Ito
Keyword(s):  
Comparison Theorem ◽  
Complex Multiplication ◽  
K3 Surfaces ◽  
Explicit Description ◽  
Brauer Group ◽  
K3 Surface ◽  
Good Reduction ◽  
Picard Number ◽  
Reduction Modulo ◽  
Modulo P

Abstract We study the good reduction modulo $p$ of $K3$ surfaces with complex multiplication. If a $K3$ surface with complex multiplication has good reduction, we calculate the Picard number and the height of the formal Brauer group of the reduction. Moreover, if the reduction is supersingular, we calculate its Artin invariant under some assumptions. Our results generalize some results of Shimada for $K3$ surfaces with Picard number $20$. Our methods rely on the main theorem of complex multiplication for $K3$ surfaces by Rizov, an explicit description of the Breuil–Kisin modules associated with Lubin–Tate characters due to Andreatta, Goren, Howard, and Madapusi Pera, and the integral comparison theorem recently established by Bhatt, Morrow, and Scholze.

scholarly journals On large Picard groups and the Hasse Principle for curves and K3 surfaces

1996 ◽  
Vol 76 (2) ◽  
pp. 165-189 ◽  
Author(s):  
Daniel Coray ◽  
Constantin Manoil
Keyword(s):  
K3 Surfaces ◽  
Hasse Principle ◽  
Picard Groups

scholarly journals The Brauer group of cubic surfaces

1993 ◽  
Vol 113 (3) ◽  
pp. 449-460 ◽  
Author(s):  
Sir Peter Swinnerton-Dyer
Keyword(s):  
Finite Group ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Rational Surface ◽  
Hasse Principle ◽  
Weak Approximation ◽  
Brauer Group ◽  
Cubic Surfaces ◽  
Image Position

1. Let V be a non-singular rational surface defined over an algebraic number field k. There is a standard conjecture that the only obstructions to the Hasse principle and to weak approximation on V are the Brauer–Manin obstructions. A prerequisite for calculating these is a knowledge of the Brauer group of V; indeed there is one such obstruction, which may however be trivial, corresponding to each element of Br V/Br k. Because k is an algebraic number field, the natural injectionis an isomorphism; so the first step in calculating the Brauer–Manin obstruction is to calculate the finite group H1 (k), Pic .

scholarly journals Upper bounds for the height of S-integral points on elliptic curves

2013 ◽  
Vol 32 (1) ◽  
pp. 125-141
Author(s):  
Vincent Bosser ◽  
Andrea Surroca
Keyword(s):  
Elliptic Curves ◽  
Upper Bounds ◽  
Integral Points

scholarly journals The Brauer–Manin obstruction for integral points on curves

2010 ◽  
Vol 149 (3) ◽  
pp. 413-421 ◽  
Author(s):  
DAVID HARARI ◽  
JOSÉ FELIPE VOLOCH
Keyword(s):  
Elliptic Curves ◽  
Negative Answer ◽  
Rational Curves ◽  
Positive Answer ◽  
Hasse Principle ◽  
Strong Version ◽  
Integral Points ◽  
Local Conditions

AbstractWe discuss the question of whether the Brauer–Manin obstruction is the only obstruction to the Hasse principle for integral points on affine hyperbolic curves. In the case of rational curves we conjecture a positive answer, we prove that this conjecture can be given several equivalent formulations and we relate it to an old conjecture of Skolem. Finally, we show that for elliptic curves minus one point a strong version of the question (describing the set of integral points by local conditions) has a negative answer.

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